mathematics

Getis–Ord statistics

Getis–Ord statistics is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Getis–Ord statistics rather than just read about it. In short: Getis–Ord statistics, also known as Gi*, are used in spatial analysis to measure the local and global spatial autocorrelation. Developed by statisticians Arthur Getis and J.

Key takeaways

Getis–Ord statistics belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Getis–Ord statistics to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Getis–Ord statistics from memory before moving on to harder problems.

Reference excerpt

Getis–Ord statistics, also known as Gi*, are used in spatial analysis to measure the local and global spatial autocorrelation. Developed by statisticians Arthur Getis and J. Keith Ord they are commonly used for Hot Spot Analysis to identify where features with high or low values are spatially clustered in a statistically significant way. Getis-Ord statistics are available in a number of software libraries such as CrimeStat, GeoDa, ArcGIS, PySAL, and R.

Local statistics

There are two different versions of the statistic, depending on whether the data point at the target location i {\displaystyle i} is included or not

G i = ∑ j ≠ i w i j x j ∑ j ≠ i x j {\displaystyle G_{i}={\frac {\sum _{j\neq i}w_{ij}x_{j}}{\sum _{j\neq i}x_{j}}}}

G i ∗ = ∑ j w i j x j ∑ j x j {\displaystyle G_{i}^{*}={\frac {\sum _{j}w_{ij}x_{j}}{\sum _{j}x_{j}}}}

Here x i {\displaystyle x_{i}} is the value observed at the i t h {\displaystyle i^{th}} spatial site and w i j {\displaystyle w_{ij}} is the spatial weight matrix which constrains which sites are connected to one another. For G i ∗ {\displaystyle G_{i}^{*}} the denominator is constant across all observations. A value larger (or smaller) than the mean suggests a hot (or cold) spot corresponding to a high-high (or low-low) cluster. Statistical significance can be estimated using analytical approximations as in the original work however in practice permutation testing is used to obtain more reliable estimates of significance for statistical inference.

Global statistics The Getis-Ord statistics of overall spatial association are

G = ∑ i j , i ≠ j w i j x i x j ∑ i j , i ≠ j x i x j {\displaystyle G={\frac {\sum _{ij,i\neq j}w_{ij}x_{i}x_{j}}{\sum _{ij,i\neq j}x_{i}x_{j}}}}

G ∗ = ∑ i j w i j x i x j ∑ i j x i x j {\displaystyle G^{*}={\frac {\sum _{ij}w_{ij}x_{i}x_{j}}{\sum _{ij}x_{i}x_{j}}}}

The local and global G ∗ {\displaystyle G^{*}} statistics are related through the weighted average

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Getis–Ord statistics

Start with the simplest possible case. Write down what Getis–Ord statistics claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Getis–Ord statistics before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Getis–Ord statistics ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Getis–Ord statistics

In research: Getis–Ord statistics appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Getis–Ord statistics in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Getis–Ord statistics is common in secondary-school and first-year university syllabi. It links to neighbouring topics Covariance and correlation, Spatial analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Getis–Ord statistics outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Getis–Ord statistics in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Getis–Ord statistics means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Getis–Ord statistics out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Getis–Ord statistics in simple terms?

Getis–Ord statistics, also known as Gi*, are used in spatial analysis to measure the local and global spatial autocorrelation. Developed by statisticians Arthur Getis and J.

Why does Getis–Ord statistics matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Getis–Ord statistics?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Getis–Ord statistics.